Finally, we shall discuss the connection with the Elekes-Rónyai theorem on expanding polynomials. If \(\phi _1,\phi _2,\phi _3\)  are smooth functions, then after a change of coordinates we can locally write \(\phi _1(x,y)=x\) , \(\phi _2(x,y)=y\) , and \(\phi _3(x,y)=P(x,y)\)  for some smooth function \(P\) . Suppose that \(A\)  is a \((\delta ,\alpha )_2\)  set, and that \(\phi _1(A)\)  and \(\phi _2(A)\)  each have size about \(|A|^{1/2}\) . If we write \(A_1=\phi _1(A)\)  and \(A_2=\phi _2(A)\) , then \(A\subset A_1\times A_2\) , and the sets \(A\)  and \(A_1\times A_2\)  have comparable size. Following the discussion above, we would like to show that \(|P(A)|\)  is much larger than \(|A|^{1/2}\) , or almost equivalently, that \(P(A_1\times A_2)\)  is much larger than \(|A|^{1/2}\) . If \(P\)  was a polynomial and if \(A_1\)  and \(A_2\)  were discrete sets of cardinality \(n\) , then Elekes and Rónyai [1]} proved that either \(P(A_1\times A_2)\)  is much larger than \(n\) , or \(P\)  has a very specific form (for example \(P(x,y) = x+y\) ). We will reduce the results discussed above about non-parallelizable \(d\) -webs to a discretized analogue of the Elekes-Rónyai theorem for smooth functions.
